# How to determine the theoretical limit on the distance estimated by stellar parallax?

To make the question a bit more realistic let the following restrictions be assumed:

• the length of the mission is 50 years (roughly 1/2 human lifetime scale i.e. 50 years so that we get 30+ years to enjoy the results of our work!)
• brightness of the stars needs to be taken into account: stars will become fainter as the distance increases
• the diameter of the telescope used is limited to D=10m
• interferometric setups are allowed

Am I missing anything?

the limits of stellar parallax are as follows.

The upper limit on the baseline length is the diameter of the earth's orbit around the sun, about 186,000 miles.

The lower limit on the subtended angle measurement is the positional resolution limit of the mechanical aiming system used in astronomical telescopes. This is approximately the width of a penny as seen at a distance of two miles.

• Thanks, but I am not limiting the measurements to ground based observations. Furthermore, the lower limit on the subtended angle is definitely not determined by the precision of the mechanical aiming of the telescope. The ultimate limit will be close to the resolution of the system. With adequate sampling this will be near the value of the resolution of the system, assuming that the SNR is adequate. – Xoxarle Sep 10 '18 at 22:29
• what I quoted- baseline and subtended angle- definitely are the limiters for optical telescope-based stellar parallax measurements. Do you want a reference? – niels nielsen Sep 10 '18 at 22:38
• Thanks, a reference will not be necessary, I will answer in two parts as the comment is too long. It is correct that a baseline length is a limitation, however, I did not specify that the telescope needs to operate at optical wavelengths, nor that it should be ground-based, or indeed Earth based. – Xoxarle Sep 10 '18 at 22:53
• Re. the subtended angle, I reiterate that the mechanical aiming of the telescope has little to do with the microarcsecond-precision parallax measurements. I claim that the limit on the measurement of the subtended angle will ultimately come from the resolving power of the system. Assuming that one has a very poor pointing precision of the telescope, the exact position can always be determined by reconstruction of the observed field of view, assuming that some of the observed sources haven't moved significantly between successive measurements. These reference sources can be e.g. quasars. – Xoxarle Sep 10 '18 at 22:55
• see the wikipedia article on this topic. – niels nielsen Sep 10 '18 at 23:17

There was a proposed NASA project called the Innovative Interstellar Explorer that could reach an estimated distance of 1000 AU after one hundred years, using a combination of gravity assist and ion thrusters. So that's 500 AU after 50 years, which in theory extends the baseline by a factor of 250, and by pure geometric reasoning the distance limit should also be extended by a factor of 250, i.e. more than two million light years. But I think it will be very difficult to accomplish this in practice, since most distant objects are quite faint.

• Yes, I agree, as the volume of the accessible space increases, the observed brightness of the stars will certainly decrease. However, we can resolve individual stars in the local group galaxies (e.g. Andromeda and Triangulum galaxies, Hubble, 2.4-m mirror), so I am guessing that the limit on the measurement of the parallax angle would be reached sooner than the brightness limit. – Xoxarle Sep 11 '18 at 13:37